cp's OEIS Frontend

A094831 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 3.

%I A094831 #32 Aug 22 2025 13:31:38
%S A094831 1,2,6,19,62,207,703,2417,8382,29242,102431,359790,1266103,4460939,
%T A094831 15730497,55500634,195890270,691566411,2441886670,8623112591,
%U A094831 30453261927,107553444913,379864424726,1341658806066,4738726458775
%N A094831 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 3.
%C A094831 In general, a(n) = (2/m)*Sum_{r=1..m-1} sin(r*j*Pi/m)*sin(r*k*Pi/m)*(2*cos(r*Pi/m))^(2n) counts (s(0), s(1), ..., s(2n)) such that 0 < s(i) < m and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = j, s(2n) = k.
%C A094831 A comparison of their recurrence relations shows that this sequence is the even bisection of A188048. - _John Blythe Dobson_, Jun 20 2015
%H A094831 Michael De Vlieger, <a href="/A094831/b094831.txt">Table of n, a(n) for n = 0..1825</a>
%H A094831 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,1).
%F A094831 a(n) = (2/9) * Sum_{r=1..8} sin(r*Pi/3)^2*(2*cos(r*Pi/9))^(2*n).
%F A094831 a(n) = 6*a(n-1) - 9*a(n-2) + a(n-3).
%F A094831 G.f.: (1-4*x+3*x^2)/(1-6*x+9*x^2-x^3).
%t A094831 CoefficientList[Series[(1 - 4 x + 3 x^2)/(1 - 6 x + 9 x^2 - x^3), {x, 0, 24}], x] (* _Michael De Vlieger_, Feb 12 2022 *)
%o A094831 (PARI) Vec((1-4*x+3*x^2)/(1-6*x+9*x^2-x^3) + O(x^30)) \\ _Michel Marcus_, Jun 21 2015
%Y A094831 Cf. A188048.
%K A094831 nonn,easy,changed
%O A094831 0,2
%A A094831 _Herbert Kociemba_, Jun 13 2004